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1 定理的来源等腰三角形两底角的平分线相等,这是每个初中学生都能证明的命题.而它的逆命题:两条内角平分线相等的三角形是等腰三角形,却是一道脍炙人口的几何难题.这个命题是雷米欧斯(Lehmus)于1840年给瑞士著名数学家斯图姆(Sturm)的一封信中提出的,并请求给出一个纯几何的证明,而斯图姆又将问题提供给一些数学家.当时德国的几何学权威斯坦纳(Steiner)首先给出了它的证明,此后该命题就以斯坦纳——雷米欧斯定理而闻名于世.一百多年来,这个定理引起了众多数学爱好者的兴趣,得出了一个个精妙的证明和各种推广.本文对此定理进行了新的推广,得到几个结论优美、证法独特的新命题,它们的证明依赖于下面的引理.
1 The Origin of the Theorem The bisecting lines of the isosceles triangles are equal. This is a proposition that every junior high school student can prove. In contrast to its proposition, the triangle whose two internal angle bisectors are equal is an isosceles triangle, but it is a popular population. The geometric puzzle. This proposition was put forward by Lehmus in a letter to the famous Swiss mathematician Sturm in 1840, and requested to give a proof of pure geometry, and Sturm. The question was again provided to some mathematicians. At that time, Steiner, the German authority of geometry, first gave its proofs. Since then, the proposition has been known to Steiner-Remyus Theorem. More than 100 In recent years, this theorem has attracted the interest of many mathematicians and has come up with exquisite proofs and various popularizations. This paper has made a new promotion of this theorem and has obtained several new propositions with beautiful conclusions and unique proofs. Their proof depends on the following lemmas.